On the extinction of measure-valued critical branching Brownian motion
type: Item
instance of: scholarly article
title: On the extinction of measure-valued critical branching Brownian motion
zbMATH Open document ID: 0694.60043
author: Li Liu
author: Carl Mueller
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: Let \((X_ t)\) be the measure-valued branching motion on \(R^ d\) satisfying \[ E^{\mu}[\exp \{-\int \phi (x)dX_ t(x)\}]=\exp \{-\int u_{\phi}(t,x)\mu (dx)\}, \] where \(X_ 0\equiv \mu\) and \(u_{\phi}\) satisfies the initial value problem \((\partial /\partial t)u=\Delta u- u^ 2\), \(u(0,\cdot)=\phi\). Let \(\rho_ t\) be the diameter of the support of \(X_ t\), and let \(\xi =\inf \{t: X_ t(R^ d)=0\}\) be the extinction time of the process. The authors give an elementary, straightforward proof that if \(\mu\) has compact support, then \(\lim_{t\uparrow \xi}\rho_ t=0\) \(P^{\mu}\)-almost surely.
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60F99
zbMATH DE Number: 4137115
zbMATH Keywords: random measure
zbMATH Keywords: branching Brownian motion
zbMATH Keywords: measure-valued branching motion
zbMATH Keywords: extinction time
zbMATH Keywords: compact support
reviewed by: Alan F. Karr
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991166
OpenAlex ID: W1967408480
DOI: 10.1214/AOP/1176991166
description: scientific article; zbMATH DE number 4137115